We study the operation E → cl (E) defined on subsets E of a unital ring R, where x ∈ cl (E) if (x + Rb) ∩ E ≠ ∅ for each b in R such that Rx + Rb = R. This operation, which strongly resembles a closure, originates in algebraic K-theory. For any left ideal L we show that cl (L) equals the intersection of the maximal left ideals of R containing L. Moreover, cl (Re) = Re + rad (R) if e is an idempotent in R, and cl (I) = I for a two-sided ideal I precisely when I is semi-primitive in R (i.e. rad (R/I) = 0). We then explore a special class of von Neumann regular elements in R, called persistently regular and characterized by forming an "open" subset Rpr in R, i.e. cl (R\Rpr) = R\Rpr. In fact, R\Rpr = cl (R\Rr), so that Rpr is the "algebraic interior" of the set Rr of regular elements. We show that a regular element x with partial inverse y is persistently regular, if and only if the skew corner (1 - xy)R(1 - yx) is contained in Rr. If I reg (R) denotes the maximal regular ideal in R and [Formula: see text] the set of quasi-invertible elements, defined and studied in [4], we prove that [Formula: see text]. Specializing to C*-algebras we prove that cl (E) coincides with the norm closure of E, when E is one of the five interesting sets R-1, [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text], and that Rpr coincides with the topological interior of Rr. We also show that the operation cl respects boundedness, self-adjointness and positivity.